Name: _______________________
Class: __________
Date:_____________
Math SL: 19C, 20A Derivatives of ln x
and trig functions
Warm-up
1. Given q = p , evaluate the following:
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a.
cos2q
b.
cosq 2
c.
(cosq )2
2. Two functions f and g are defined as follows:
f (x) = cos x, 0  x  2;
g (x) = 2x + 1, x  .
Solve the equation (g  f)(x) = 0.
(Topic 2: Functions #4)
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19C: Derivatives of Logarithmic Functions
Today’s Objectives:
1. to introduce the derivatives of logarithmic functions
2. to introduce the derivatives of sin x, cos x, and tan x
3. to use the chain rule to find the derivatives of sin[f(x)], cos[f(x)] and tan[f(x)]
Derivatives of Logarithmic Functions
dy 1
=
dx x
We can use the chain rule to show that:
If y = ln x then
If y = ln f (x) then
Derivative Practice:
1. f (x) = ln x 2
3.
f (x) = ln ( 2x 4 - 5x2 )
dy f ¢(x)
=
dx
f (x)
2.
f (x) = ln (1- 3x 2 )
4.
f (x)= ln 2 - ( x +1)8
x
Note: This is how it appears in your information booklet.
f (x) = ln x Þ f ¢(x) =
2
1
x
20A The Derivatives of the Basic Trig Functions
We can summarize the derivatives of trigonometric functions in the following
table:
Function f (x)
Derivative f ¢(x)
sin x
cos x
cos x
-sin x
tan x
1
cos 2 x
Note: The above derivatives can be found in your information booklets, but
slightly different.
The Derivatives of sin[ f (x)] , cos[ f (x)] and tan[ f (x)]
We can use the chain rule to find these derivatives. They are summarized in the
following table:
Function
Derivative
sin[ f (x)]
cos[ f (x)] f ¢(x)
cos[ f (x)]
-sin[ f (x)] f ¢(x)
tan[ f (x)]
sec2 [ f (x)] ´ f ¢(x) =
f ¢(x)
cos 2 [ f (x)]
Note: The above the derivatives that involve the chain rule are NOT in your
information booklets.
Determine the derivative functions of the following:
1. f(x) = cos (2x+1)
2. f(x) = sin (x2 – 4)
æ
1ö
3. f(x) = tan ç 4x 3 - ÷
xø
è
3
Hmwk #8
19C Derivatives of Logarithmic Functions
pg.562 # 1-3 every second question
20A Derivatives of Trig Functions
pg.574 # 1-3 every second question, 4, 5
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